https://file.aiquickdraw.com/mathbot/file/1729081070871ubmlt5h4.jpg

Let's go step by step through the different parts of this question.

---

### Part (a) - State the hypotheses we are interested in testing.
We are comparing the **winning ability** of the Dallas Cowboys when Tony Romo versus Brad Johnson was the starting quarterback. 

- **Null hypothesis (H₀):** The Cowboys have the same ability to win with either quarterback.  
  \[ H_0: p_{\text{Romo}} = p_{\text{Johnson}} \]
- **Alternative hypothesis (H₁):** The Cowboys have a greater ability to win with Romo.  
  \[ H_1: p_{\text{Romo}} > p_{\text{Johnson}} \]

---

### Part (b) - Summarize the Cowboys' performances with a two-way table.
From the problem:
- Tony Romo: 8 wins and 5 losses (13 games)
- Brad Johnson: 1 win and 2 losses (3 games)

Here is the **two-way table:**

| **Quarterback** | **Wins** | **Losses** | **Total Games** |
|-----------------|----------|------------|-----------------|
| Tony Romo       | 8        | 5          | 13              |
| Brad Johnson    | 1        | 2          | 3               |
| **Total**       | 9        | 7          | 16              |

---

### Part (c) - Explain how to use note cards to simulate the difference in winning percentage.
The goal of this simulation is to assume that the Cowboys' ability to win is **the same** regardless of the quarterback (i.e., under the null hypothesis).

**Steps for the simulation:**
1. Write 9 "Win" cards and 7 "Loss" cards to represent the outcomes of the 16 games.
2. Shuffle the cards thoroughly and randomly assign 13 cards to represent Romo's games and 3 cards for Johnson's games.
3. Calculate the winning percentage for both Romo and Johnson based on the assigned cards.
4. Compute the **difference in winning percentage**:  
   \[ \text{Simulated Difference} = p_{\text{Romo}} - p_{\text{Johnson}} \]
5. Repeat this process multiple times (at least 25 times) to build a distribution of the simulated differences.

---

### Part (d) - Estimate the p-value from the dotplot.
The **p-value** is the probability of observing a difference in winning percentage as extreme as or more extreme than the observed difference (28.2%), assuming the null hypothesis is true.

From the dotplot:
- The observed difference was **28.2%**.
- In the dotplot, there are very few points at or beyond 28.2% (positive side).
- Let's say there are 2 dots at or beyond this threshold out of 25 trials.

The **p-value** estimate is:
\[
p\text{-value} = \frac{\text{Number of simulations with difference } \geq 28.2\%}{\text{Total number of simulations}}
= \frac{2}{25} = 0.08
\]

---

### Part (e) - Conclusion based on the p-value.
If the **p-value = 0.08**, it means that under the null hypothesis, there is an 8% chance of observing a difference in winning percentage as large as or larger than 28.2%.

- If we use a significance level of **0.05**, the p-value is **not small enough** to reject the null hypothesis.
- **Conclusion:** There is not enough evidence to conclude that the Cowboys had a greater ability to win with Tony Romo compared to Brad Johnson.

---

### Follow-up questions:
1. What if the p-value was smaller than 0.05? How would the conclusion change?
2. How many simulations would increase the accuracy of the p-value estimate?
3. Could we improve the simulation process in any way?
4. How does the choice of quarterback influence the interpretation of these results?
5. What other statistical methods could we use for similar comparisons?

---

### Tip:
When dealing with small sample sizes, as in Brad Johnson's 3 games, p-values may not always give the most reliable conclusions. Bootstrap methods or confidence intervals could provide additional insights.

In the 2008 NFL season, the Dallas Cowboys had 9 wins and 7 losses with two different starting quarterbacks. Tony Romo had a winning percentage of 61.5%, and Brad Johnson had a winning percentage of 33.3%. Does the difference in winning percentage provide convincing evidence that the Cowboys had a greater ability to win with Romo? The task involves stating hypotheses, summarizing performances, using simulations, and interpreting a p-value from a dotplot.

Explore hypothesis testing with a real-world NFL case comparing Tony Romo and Brad Johnson's winning percentages from the 2008 season. Learn how to calculate the p-value, use simulation-based inference, and analyze results to determine if Romo had a higher ability to win.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation